Hey all,
I'm glad to read all of these answers. This is a valid question, and deserves some valid answers, not just "here comes another tone thread" responses. We've mostly heard opinions, and of course, they get old.
Here are some facts that have been missing to date in any discussion about timbre and vibrations etc.
1. A vibrating string is governed by the WAVE equation, a second order partial differential equation, typically of the form:
ytt = c2 * yxx - b * yt
where y is the displacement of the string (from the starting position), and changes with time and along the length of the string. ytt is the second derivative with respect to time, and yxx is the second derivative with respect to position along the length of the string. The last term is a decay term due to the transfer of energy to the ends of the string, which for a stringed instrument is the bridge & frets or nut (if an open string), and shows how the displacement of the string (i.e. its lateral movement) changes with time. A related acoustical term is the "decay" of the note - these are not the same - the acoustical decay is the sound resulting from the dampening of the string & energy transfer to the string ends.
2. Next, there is a question of the ENERGY TRANSFER within the instrument hardware. Because vibrational energy transfers differently in different materials, the decay term in the above-mentioned differential equation will change with each material, and because an instrument is a combination of several materials (metal bridge, body / fingerboard wood, nut, frets, etc.), the decay coefficient is very likely a highly nonlinear function to account for the impact of each individual component, but will ultimatelly depend on the nature of vibration propagation within that material.
3. So, there is a transfer of energy from the ends of the string into the bridge / fret & subsequently into the neck wood & body wood, and the vibration of the string is affected. Since the vibration of the string causes a change in the magnetic field (measured by the pickup), the electrical signal from the pickup magnet changes, and after appropriate amplification, the sound is heard. In the end, the decay coefficient in the differential equation determines how the vibrating string changes, and since the decay coefficient is based on the materials into which the energy propagates, that material (i.e. wood, bridge material, etc.) affects the string vibration and dampening, which affect the acoustical properties of the vibrating string, i.e. the timbre, or in crude terminology, the 'tone'. In an acoustical instrument where the wood creates the air vibrations directly, there is more of an effect, i.e., the transfer of vibrational energy from the string into the wood is direct, instead of being transfered from string to bridge to wood or string to fret / nut to wood.
NOTE: This nice analytical differential equation represents the WAVE equation in a very basic form, and is not the exact governing equation for a plucked string in a realistic situation. The exact form would look similar to this, but this is an approximation, where the real governing equation is FAR FAR more complex & probably highly nonlinear.
4. So, knowing that there theoretically is a difference, the question becomes how we can quantify the influence of a given piece of wood. If we were able to make a nice, compact governing equation for the vibration of an electric bass string, we'd have to quantify the vibration transfer between each subcomponent in the system. (Head spinning yet?) Every single other variable would have to be fixed, which is also almost exactly impossible. You'd have to have a plucking machine (to generate the same force of pluck each time), identical strings, bridge, etc. This isn't happening in your living room or even in a luthier's shop, I can assure you. To definitely fix all of these variables is not a trivial issue. A secondary method for quantification would be to model the waveforms of the string decay under different conditions, and empirically determine a difference between the two based on the string decay. This method is less than ideal, unless extreme care was taken to utilize precisely identical electrical conditions.
Everyone is correct in this thread. Wood WILL affect the timbre of a vibrating string through the transfer of vibrational energy. It IS, however, impossible to quantify a difference through the use of only one variable (wood type) while isolating and fixing all remaining variables variables. Examples of those variables include wood type, wood moisture content, wood porosity structure, structural shapes, bridge material, bridge shape, bridge to body contact, fret shape, fret material, nut shape, nut material, fret and nut to wood contact, pickup characteristics, etc.etc.) Since two pieces of the same wood species are different, otherwise identical basses are NOT identical, thus negating any type of worthwhile comparison.
If I ever hit the lottery, I'm going to establish a bass research institute & quantifiably determine all of these arguments in a rigorous, scientific fashion. Until then, we're stuck with our ears & what sounds good to us.
imp
P.S. I know my answer is correct - in fact, I challenge anyone reading this to prove my argument wrong.